| L(s) = 1 | − 2·2-s − 4-s + 2·8-s − 8·11-s + 3·16-s + 16·22-s − 4·23-s − 9·25-s − 22·29-s + 2·32-s − 6·37-s + 6·43-s + 8·44-s + 8·46-s + 18·50-s − 28·53-s + 44·58-s − 14·64-s − 38·71-s + 12·74-s − 6·79-s − 12·86-s − 16·88-s + 4·92-s + 9·100-s + 56·106-s − 26·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 0.707·8-s − 2.41·11-s + 3/4·16-s + 3.41·22-s − 0.834·23-s − 9/5·25-s − 4.08·29-s + 0.353·32-s − 0.986·37-s + 0.914·43-s + 1.20·44-s + 1.17·46-s + 2.54·50-s − 3.84·53-s + 5.77·58-s − 7/4·64-s − 4.50·71-s + 1.39·74-s − 0.675·79-s − 1.29·86-s − 1.70·88-s + 0.417·92-s + 9/10·100-s + 5.43·106-s − 2.51·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + T + p T^{2} + 3 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + 9 T^{2} + 63 T^{4} + 349 T^{6} + 63 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 4 T + 32 T^{2} + 87 T^{3} + 32 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 3 p T^{2} + 66 p T^{4} + 13439 T^{6} + 66 p^{3} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 18 T^{2} + 216 T^{4} + 2797 T^{6} + 216 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 39 T^{2} + 1011 T^{4} + 20009 T^{6} + 1011 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 2 T + 44 T^{2} + 33 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 11 T + 101 T^{2} + 549 T^{3} + 101 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 57 T^{2} + 3927 T^{4} + 3731 p T^{6} + 3927 p^{2} T^{8} + 57 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 3 T + 87 T^{2} + 249 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 84 T^{2} + 4722 T^{4} + 236077 T^{6} + 4722 p^{2} T^{8} + 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 3 T + 105 T^{2} - 285 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 99 T^{2} + 9315 T^{4} + 451393 T^{6} + 9315 p^{2} T^{8} + 99 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 14 T + 170 T^{2} + 1221 T^{3} + 170 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 171 T^{2} + 18477 T^{4} + 1250773 T^{6} + 18477 p^{2} T^{8} + 171 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 + 102 T^{2} + 12918 T^{4} + 712865 T^{6} + 12918 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 90 T^{2} + 353 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 19 T + 329 T^{2} + 2925 T^{3} + 329 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 363 T^{2} + 59331 T^{4} + 5567537 T^{6} + 59331 p^{2} T^{8} + 363 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 3 T + 159 T^{2} + 367 T^{3} + 159 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 270 T^{2} + 41004 T^{4} + 4002649 T^{6} + 41004 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 288 T^{2} + 36990 T^{4} + 3427693 T^{6} + 36990 p^{2} T^{8} + 288 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 471 T^{2} + 100065 T^{4} + 12364877 T^{6} + 100065 p^{2} T^{8} + 471 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.93954988432225948446276040877, −4.46552232819000247294215086599, −4.38781885962878232526144501644, −4.30812590717530669356106228730, −4.23151368857614199698505371900, −4.18506651682260925415017126064, −3.94109110643597913273445801217, −3.68760569405945060636780453374, −3.61844007154990117228613320581, −3.39695274932604425468309297275, −3.38703258345505784608521168007, −3.36998975564205608437255943129, −2.82765456779840151515092627696, −2.75747983054577254754948720267, −2.75746451716673335470451582018, −2.66632637025908365710493639385, −2.37981899754266625134904213432, −2.15190795374646663747122344116, −2.10158919502305365507751606798, −1.79685367213301580248342426195, −1.64899399266191682985086320293, −1.44744752144681903920795338620, −1.41176781581954927138663295568, −1.02320146267459245981160774630, −0.975693144638551746428826892199, 0, 0, 0, 0, 0, 0,
0.975693144638551746428826892199, 1.02320146267459245981160774630, 1.41176781581954927138663295568, 1.44744752144681903920795338620, 1.64899399266191682985086320293, 1.79685367213301580248342426195, 2.10158919502305365507751606798, 2.15190795374646663747122344116, 2.37981899754266625134904213432, 2.66632637025908365710493639385, 2.75746451716673335470451582018, 2.75747983054577254754948720267, 2.82765456779840151515092627696, 3.36998975564205608437255943129, 3.38703258345505784608521168007, 3.39695274932604425468309297275, 3.61844007154990117228613320581, 3.68760569405945060636780453374, 3.94109110643597913273445801217, 4.18506651682260925415017126064, 4.23151368857614199698505371900, 4.30812590717530669356106228730, 4.38781885962878232526144501644, 4.46552232819000247294215086599, 4.93954988432225948446276040877
Plot not available for L-functions of degree greater than 10.
